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The dimensions of Hausdorff and Mendes France. A comparative study

R. Hansen·M. Piacquadio
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Abstract

This paper contains a comparative study of two families of simple curves drawn in the plane. On the one hand, we have the fractal curves on the unit interval, with self-similar structure, which have associated a Hausdorff dimension. On the other hand, we have the opposite: a class of locally rectifiable unbounded curves, which have another "fractional dimension" defined by M. Mendes France. We propose a geometrical constructive process that will allow us to obtain - as the limit of a sequence of polygonal curves - one curve of the first family, by contractive transformations; and another of the second family, by expansive transformations. Thanks to this process of linking curves from both families, we are able to compare their dimensions - our aim in this work -, and to obtain interesting results such as the equality of the latter in the case of strict self-similarity.

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BibTeXRIS

R. Hansen, M. Piacquadio. 2000-04-10. The dimensions of Hausdorff and Mendes France. A comparative study. https://arxiv.org/abs/math/0004060

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